Hence x+y is the greatest common measure sought, and (3.) Required the greatest common measure of the two polynomials Here 6a6a2y+2ay2—2y3 = 2 (3a3—3a2y+ay2—y3) And therefore, by suppressing the factors 2 and 3, which have no common measure, we have to find the greatest common measure of 3a3-3a2y+ay2—y3 and 4a2—5ay+y2. 4a2-5ay+y2) 3a3— 3a2y+ ay2— y3 Hence a―y is the greatest common measure of the polynomials a and 6. (4.) Required the greatest common measure of the terms of the fraction a®—a2x1 Here a is a simple factor of the numerator, and a' is a factor of the denominator; hence a2 is the greatest common measure of these simple factors, which must be reserved to be introduced into the greatest common measure of the other factors of the terms of the proposed fractions; viz.: Therefore a2 (a2—x2) is the greatest common measure; and hence (1.) Find the greatest common measure of 2a2x2, 4x2y2, and 6x3y. (2.) Find the greatest common measure of the two polynomials a3—a2b+ Sab-363, and a2-5ab+4b2. (3.) What is the greatest common measure of x3-xy2 and x2+2xy+y2? 13 (4.) Find the greatest common measure of æ3· (b+c) x3—b (2b+c) x2+b3x. . . . . (b). (6.) Find the greatest common measure of the polynomials (a) 32. We have already defined a multiple of a quantity to be any quantity that contains it exactly; and a common multiple of two or more quantities is & quantity that contains each of them exactly. The least common multiple, of two or more quantities, is therefore the least quantity that contains each of them exactly. 33. To find the least common multiple of two quantities. Divide the product of the two proposed quantities by their greatest common measure, and the quotient is the least common multiple of these quantities; or divide one of the quantities by their greatest cominon measure, and multiply the quotient by the other. Let a and b be two quantities, d their greatest common measure; and m their least common multiple; then let a= hd, and b = kd; and since d is the greatest common measure, h and k can have no common factor, and hence their least common multiple is hk; therefore hkd is the least common multiple of hd and k d; hence, m=hkd= hk d2 hdxkd ахь ab d = = d Q. E. D. EXAMPLES. (1.) Find the least common multiple of 2a2x and Sa3x3. (2.) Find the least common multiple of 4x2 (x2—y2) and 12x3 (x3—y3). Here d= 4x2 (x—y), and therefore we have ab m= .: = 4x2 (x2—y2) × 12x3 (x'—y') = 12x2(x+y) (x'—y'); 4x (x-y) or m = 12x7+12x°y—12x*y3—12x3y'. (3.) Find the least common multiple of x2+2xy+y2 and x3—x y2. Here dx+y, and therefore we get a x+2xy+y2 m= d x+y = (x+y) (x3-x y2) = x(x+y) (x2—y2) : (4.) What is the least common multiple of x-5x3 + 9x2 — 7x+2, and 2--6x2+8x-3? By the process for finding the greatest common measure, we find = x3—2x1—6x3+20x2-19x+6, the least common multiple. (5.) Find the least common multiple of a2-2ab+b2, and aa—b1. (6.) Find the least common multiple of a2—b2, and a3+b3. (7.) Find the least common multiple of x2—y2, and x3—y3. (8.) Find the least common multiple of y2-8y+7, and y2+7y-8. (1.) (a-b) (a1—b1). (2.) (a-b) (a3+b3). ANSWERS. (3.) (x+y) (x3—y3) 34. Every common multiple of two quantities, a and b, is a multiple of m, the least common multiple. For let m' be a common multiple of a and b, then, because m' is greater than m; and if we suppose that m' is not a multiple of m, we have, as in the annexed scheme, Now the remainder k is always less than m the divisor; hence, since a and b measure m and m', it is evident by (2) that a and b measure m'—h m, or k; therefore k is a common multiple of a and b, and it has been proved to be less than m, the least common multiple, which is absurd; hence m must measure m', or m' is a multiple of m. 35. To find the least common multiple of three or more quantities. Let a, b, c, d, &c., be the proposed quantities; find m the least common multiple of a and b c and m d and m' Then, since every multiple of a and b measures m, their least common muliple, the quantity sought, z, measures m; but x also measures c; therefore measures both c and m, and thence it measures m'; but a measures d and m', and therefore must measure m"; hence x cannot be less than m", and therefore m" is the least common multiple. EXAMPLES. 1.) Find the least common multiple of 2a2, 4a3 b2, and 6a b3. Here taking 2a2 and 4a3 b2, we find d = 2a2, and, therefore, ab 2a2 × 4a3 b2 m= ༤/、 = 2a2 = 4a3 bo. Again, taking m, or 4a3 b2, and 6a b3, we find d = 2a b2; hence m'= n = cm = 6a b3 × 4a3 b2 2ab2 12a3b3 answer required. (2.) Find the least common multiple of a-x, a2-x2, and q3—x3. Taking a-x and a2-x2, we have d=a-x; and hence m= ab a-x = x (a2x2)=a2—x. d a-x Again, taking a2—x2 and a3—x3, we find d=a—x; hence m= cm (a3-x3) (a2-x2) = a-x = (a+x)(a3—x3)= answer sougnt. (3.) Find the least common multiple of 15a2 b2, 12a b3, and 6a3 b. (4.) Find the least common multiple of 6a2 x2 (a−x), 8x3 (a2—x2) and 12 (a-x)2. (5.) Find the least common multiple of x3-x2y—xy2+y3, x3—x2y+xy2—y'; and x^—y^. (6.) Find the least common multiple of (a+b)2, (a2—b2), (a—b)2, and a2+3a2b+3a b2+b3. ANSWERS. (3.) 60ab3. (5.) x-xy-x^y+y3. (4.) 24a2 x' (a-x)2 (6.) (a+b) (a2-b2). OF ALGEBRAIC FRACTIONS. 36. Algebraic fractions differ in no respect from arithmetical fractions; and all the observations which we have made upon the latter, apply equally to the former. We shall therefore merely repeat the rules already deduced, adding a few examples of the application of each. It may be proper to remind the reader, that all our operations with regard to fractions were founded upon the three following principles: 1. In order to multiply a fraction by any number, we must multiply the numerator, or divide the denominator of the fraction by that number. 2. In order to divide a fraction by any number, we must divide the numerator, or multiply the denominator of the fraction by that number. 3. The value of a fraction is not changed, if we multiply or divide both the numerator and denominator by the same number.* REDUCTION OF FRACTIONS. 1. To reduce a fraction to its lowest terms. 37. RULE.-Divide both numerator and denominator by their greatest common measure, and the result will be the fraction in its lowest terms. When the numerator and denominator are, one or both of them, monomials, their greatest common factor is immediately detected by inspection; thus. a2bXc с in its lowest terms. 5a2b2a2bx5b5b a2bc = So also, ах in its lowest terms. ax2 ххах = If, however, both numerator and denominator are polynomials, we must nave recourse to the method of finding the greatest common measure of two algebraic quantities, developed in a former article. Thus, let it be required to reduce the following fraction to its lowest terms: 6a3-6a2y+2ay2-2y3 • These principles will be obvious from the following considerations: 1. If the numerator of a fraction be increased any number of times, the fraction itself will be Increased as many times; and if the denominator be diminished any number of times, the fraction must still be increased as many times. 2. If the denominator of a fraction be increased any number of times, or the numerator diminished the same number of times, the fraction itself will in either case be diminished the same number of times. 3. If the numerator of a fraction be increased any number of times, the fraction is increased the same number of times; and if the denominator be increased as many times, the fraction is again diminished the same number of times, and must therefore have its original value. |